Busy period, time of the first loss of a customer and the number of customers in $ M^{\varkappa}|G^{\delta}|1|B$

TL;DR

This paper analyzes the $ M^{\varkappa}|G^{\delta}|1|B$ queueing system by solving a two-sided exit problem for a combined stochastic process, deriving explicit formulas for busy periods and customer counts.

Contribution

It provides the first closed-form expressions for busy period distribution and customer numbers in the $ M^{\varkappa}|G^{\delta}|1|B$ system using resolvent sequences.

Findings

Explicit Laplace transforms for first exit time, overshoot, and process value.

Closed-form formulas for busy period distribution.

Distribution of customer numbers in transient and stationary regimes.

Abstract

A two-sided exit problem is solved for a difference of a compound Poisson process and a compound renewal process. More precisely, the Laplace transforms of the joint distribution of the first exit time, the value of the overshoot and the value of a linear component at this instant are found. Further, we study the process reflected in its supremum. We determine the main two-boundary characteristics of the process reflected in its supremum. These results are then applied for studying the $ M^{\varkappa}|G^{\delta}|1|B$ system. We derive the distribution of a busy period and the numbers of customers in the system in transient and stationary regimes. The advantage is that these results are in a closed form, in terms of resolvent sequences of the process.